Suppose circles C and C' (not necessarily the same radius) intersect in two different points. Prove that if the tangents to the two circles are perpendicular at one point of intersection, then the tangents to the two circles are also perpendicular at the second point of intersection.

A batter of neglectible height can hit a ball with an initial speed at any angle. How high should a domed stadium be at a distance x from the batter in order to prevent the ball from reaching the dome? (Neglect air resistance.)

Consider the sulfite ion. Draw the three best resonance structures. Draw and name the molecular shape. Calculate the formal charge on the sulfur atom.

A particle is moving in the plane in such a way that its coordinates (x,y) after t hours have elapsed on Tom's watch are given by x=f(t) and y=g(t). If Tom's watch gains 11 seconds per hour and Bill's watch keeps accurate time, what will be the coordinates of the particle when Bill's watch says that t minutes have elapsed?

According to the U. S. Census Bureau, the number of children per family averages 0.91 with a standard deviation of 1.11. The A. C. Nielsen Company is to randomly sample 1000 families. Nielsen would like to see at least 1000 children in the sample in order to get good information on children's TV programs. Do they have a good chance of seeing at least 1000 children in a sample of 1000 families?

Gauss's Law is given by where the basic force law is and the electric field, for a charged particle is . The force law for gravity is .

	a.	Find Gauss's Law for a gravitational field.
	b.	Assume a uniform thin plane with surface density of mass M kg/m2 exists over all space. Find the gravitational field outside the plane.

Let N be a power of 2. Prove that N cannot be represented as the sum of two or more consecutive positive integers.

Monarch butterflies are famous throughout the world for their long migrations in North America, but they are also known to be highly unpalatable. For decades naturalists have observed that the butterflies' predators reject them as food items. Curious entomologists have tasted the insects themselves and agree that, on the basis of taste, Monarchs don't belong on their menu. It has also been long known that Monarchs feed almost exclusively as larvae on members of the milkweed family, the Asclepiadaceae, also known for its bitter taste and highly poisonous cardiac glycoside compounds. Suspecting that the distasteful principle of the Monarch may be directly connected to its larval host plant, state two hypotheses that could explain the observations reported above and then design an experimental and/or analytical procedure to discriminate between the two hypotheses you stated. State how various experimental outcomes allow this discrimination.